Coset relation algebras

A measurable relation algebra is a relation algebra in which the identity element is a sum of atoms that can be measured in the sense that the “size” of each such atom can be defined in an intuitive and reasonable way (within the framework of the first-order theory of relation algebras). A large class of examples of such algebras, using systems of groups and coordinated systems of isomorphisms between quotients of the groups, has been constructed. This class of group relation algebras is not large enough to exhaust the class of all measurable relation algebras. In the present article, the class of examples of measurable relation algebras is considerably extended by adding one more ingredient to the mix: systems of cosets that are used to “shift” the operation of relative multiplication. It is shown that, under certain additional hypotheses on the system of cosets, each such coset relation algebra with a shifted operation of relative multiplication is an example of a measurable relation algebra. We also show that the class of coset relation algebras does contain examples that are not representable as set relation algebras. In later articles, it is shown that the class of coset relation algebras is adequate to the task of describing all measurable relation algebras in the sense that every atomic measurable relation algebra is essentially isomorphic to a coset relation algebra, and the class of group relation algebras is similarly adequate to the task of representing all measurable relation algebras in which the associated groups are finite and cyclic.     

On taking subalgebras of relativized relation algebras

We show that the class of all relativized (in the usual sense) relation algebras is not closed under taking subalgebras. Moreover, we show that there is a complete and atomic subalgebra which is not in the original class (relation composition is not completely additive in it). These results are in contrast with results in Maddux [82] and in Henkin-Resek [75].Presented by Jan Mycielski.Research supported by Hungarian National Foundation for Scientific Research grant No. 1810.     

Matrix relation algebras

Square matrices over a relation algebra are relation algebras in a natural way. We show that for fixed n, these algebras can be characterized as reducts of some richer kind of algebra. Hence for fixed n, the class of n × n matrix relation algebras has a first–order characterization. As a consequence, homomorphic images and proper extensions of matrix relation algebras are isomorphic to matrix relation algebras. Received July 18, 2001; accepted in final form April 24, 2002.     

Weakly representable but not representable relation algebras

We prove the existence of non-representable relation algebras the union and complementation free reducts of which can be represented, i.e. which are weakly representable. This answers Problem 3 in Jónsson [4], and has consequences concerning the complexity of the equational theory of representable relation algebras.Presented by B. Jonsson.Research supported by Hungarian National Foundation for Scientific Research grants No. 1911 and No. T7255.     

Axiomatization of identity-free equations valid in relation algebras

A finite axiom set for the identity-free equations valid in relation algebras is given. This is a simplification of the one given by Jónsson, and confirms a conjecture of Tarski. An axiom set for the identity-free equations valid in the representable relation algebras is given, too. We show that in the class of representable relation algebras, both the operation of taking converse and the identity constant are finitely axiomatizable (over the rest of the operations).Dedicated to the memory of Alan DayPresented by J. Sichler.     

Relation algebras as expanded FL-algebras

This paper studies generalizations of relation algebras to residuated lattices with a unary De Morgan operation. Several new examples of such algebras are presented, and it is shown that many basic results on relation algebras hold in this wider setting. The variety qRA of quasi relation algebras is defined and shown to be a conservative expansion of involutive FL-algebras. Our main result is that equations in qRA and several of its subvarieties can be decided by a Gentzen system, and that these varieties are generated by their finite members.     

Koszul Algebras and Their Ideals

We study associative graded algebras that have a “complete flag” of cyclic modules with linear free resolutions, i.e., algebras over which there exist cyclic Koszul modules with any possible number of relations (from zero to the number of generators of the algebra). Commutative algebras with this property were studied in several papers by Conca and others. Here we present a noncommutative version of their construction.We introduce and study the notion of Koszul filtration in a noncommutative algebra and examine its connections with Koszul algebras and algebras with quadratic Grobner bases. We consider several examples, including monomial algebras, initially Koszul algebras, generic algebras, and algebras with one quadratic relation. It is shown that every algebra with a Koszul filtration has a rational Hilbert series.__________Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 39, No. 2, pp. 47–60, 2005Original Russian Text Copyright © by D. I. PiontkovskiiSupported in part by the Russian Foundation for Basis Research under project 02-01-00468.     

Deformation of isolated hypersurface singularities and their moduli algebras

By the using of determinantal varieties from moduli algebras of hypersurface singularites the relation of the deformation of hypersurface singularities and the deformation of their moduli algebras is studied. For a type of hypersurface singularities a weak Torelli type result is proved. This weak Torelli type result showes that for families of hypersurface singularities the moduli algebras can be used to distinguish the complex structures of singularities at least in some weak sence. Research supported by NNSF     

Algebras Defined from Ordered Sets and the Varieties they Generate

We investigate ways of representing ordered sets as algebras and how the order relation is reflected in the algebraic properties of the variety (equational class) generated by these algebras. In particular we consider two different but related methods for constructing an algebra with one binary operation from an arbitrary ordered set with a top element. The two varieties generated by all these algebras are shown to be well-behaved in that they are locally finite, finitely based, and have an equationally definable order relation. We exhibit a bijection between the subdirectly irreducible algebras in each variety and the class of all ordered sets with top element. We determine the structure and cardinality of the free algebra on n-free generators and provide sharp bounds on the number of n-generated algebras in each variety. These enumeration results involve the number of quasi-orders on an n-element set.     

An algebraic dependence over the quasi-centre

Summary The authors study a class of rings generalizing both ξ-rings and rings which are (1) quasi-radical over their normalizer. The results are quite complete in the Noetherian case. In application several conditions on rings are shown to force the right duo condition. In particular it is shown that ξ-rings, algebras over the rationals satisfying (1), and algebraic algebras having no non central nilpotent elements (or a more general condition) are right duo or duo rings. Entrata in Redazione il 28 luglio 1972. This research has been supported by grant A7876 of the National Research Council of Canada. This research has been supported by grant A7877 of the National Research Council of Canada.     

Relativized relation algebras

Relativization is one of the central topics in the study of algebras of relations (i.e. relation and cylindric algebras). Relativized representable relation algebras behave much nicer than the original class RAA: for instance, one obtains finite axiomatizability, decidability and amalgamation by relativization. The properties of the class obtained by relativizing RRA depend on the kind of element with which the algebras are relativized. We give a systematic account of all interesting choices of relativizing RRA, and show that relativizing with transitive elements forms the borderline where all above mentioned three properties switch from negative to positive. Received January 24, 1993; accepted in final form October 7, 1998.     

Contractibility of the linear group in Banach spaces of measurable operators

We show contractibility to a point of the linear group for a wide class of symmetric spaces of measurable operators affiliated with several concrete non-atomic semifinite von Neumann algebras.Research supported by the Australian Research Council     

Spectra for solvable Lie algebras of bundle endomorphisms

The aim of the paper is to investigate spectral properties of the Lie algebras corresponding to the symmetry groups of certain flags of vector bundles over a compact space. Under natural hypotheses, such Lie algebras are solvable, being in general infinite dimensional. The spectral theory of finite-dimensional solvable Lie algebras of operators is extended to this natural class of infinite-dimensional solvable Lie algebras. The discussion uses the language of continuous fields of -algebras. The flag manifolds in -algebraic framework are naturally involved here, they providing the basic method for obtaining flags of vector bundles. Received: 8 October 2001 / Revised version: 4 February 2002 / Published online: 6 August 2002 Research supported from the contract ICA1–CT–2000–70022 with the European Commission.     

Notes on equational theories of relations

We describe explicitly the free algebras in the equational class generated by all algebras of binary relations with operations of union, composition, converse and reflexive transitive closure and neutral elements 0 (empty relation) and 1 (identity relation). We show the corresponding equational theory is decidable by reducing the problem to a question about regular sets. Similar results are given for two related equational theories.Presented by R. W. Quackenbush.Partially supported by a joint grant from the NSF and the Hungarian Academy of Sciences.Partially supported by a grant from the Hungarian National Foundation for Scientific Research and a joint grant from the NSF and the Hungarian Academy of Sciences. ^v-semirings of 1-closed regular sets. On the basis of this characterization, we conjectured that a set of equational axioms for the variety REL^v consists of equational axioms for the variety L^v and the equation (10). Recently, this conjecture has been proved in [6].     

Symmetric composition algebras over algebraic varieties

Let ${(X,\mathcal{O}_X)}$ be a locally ringed space. We investigate the structure of symmetric composition algebras over X obtained from cubic alternative algebras ${\mathcal{A}}$ over X generalizing a method first presented by J. R. Faulkner. We find examples of Okubo algebras over elliptic curves which do not have any isotopes which are octonion algebras and of an octonion algebra which is a Cayley-Dickson doubling of a quaternion algebra but does not contain any quadratic étale algebras.     

Representations of Frobenius-type triangular matrix algebras

The aim of this paper is mainly to build a new representation-theoretic realization of finite root systems through the so-called Frobenius-type triangular matrix algebras by the method of reflection functors over any field. Finally, we give an analog of APR-tilting module for this class of algebras. The major conclusions contains the known results as special cases, e.g., that for path algebras over an algebraically closed field and for path algebras with relations from symmetrizable cartan matrices. Meanwhile, it means the corresponding results for some other important classes of algebras, that is, the path algebras of quivers over Frobenius algebras and the generalized path algebras endowed by Frobenius algebras at vertices.     

A non-axiomatizability result in algebraic logic

In this paper we prove that the class of all relation algebras which are isomorphic to relation algebras of the form (L) for some semigroupL (so called semigroup relation algebras) is not first-order axiomatizable.Presented by E. Fried.     

Axiomatizability of reducts of algebras of relations

In this paper, we prove that any subreduct of the class of representable relation algebras whose similarity type includes intersection, relation composition and converse is a non-finitely axiomatizable quasivariety and that its equational theory is not finitely based. We show the same result for subreducts of the class of representable cylindric algebras of dimension at least three whose similarity types include intersection and cylindrifications. A similar result is proved for subreducts of the class of representable sequential algebras. Received October 7, 1998; accepted in final form September 10, 1999.